The problem describes a circuit with the following parameters:

• Charge ($Q$) = 8.5 C
• Two points $A$ and $B$ are given with positions:
  • $A = 2456 \text{ mm}$
  • $B = 7654 \text{ mm}$
• Current ($I$) = 15.87 A

The questions are:

1. (A) What is the resistance?
2. (B) If the voltage is halved, what is the new current?
3. (C) If the current is one-third of the initial value, what is the new voltage?

Let's break down each question and solve them:

Solution:

1. Determine the Voltage: To find the resistance in part (A), we need the voltage $V$ across points $A$ and $B$. Assuming we are working with a direct current (DC) circuit, and the positions of $A$ and $B$ imply a potential difference, we can use the relationship: $V = IR$ However, the voltage itself is not provided directly. Given the data, let’s proceed assuming that further details about the voltage calculation are implied by the context (such as the distance between $A$ and $B$).

2. Part (A) – Find the Resistance $R$: Once we know the voltage $V$, the resistance $R$ can be calculated using Ohm’s Law: $R = \frac{V}{I}$

3. Part (B) – Halved Voltage and New Current: If the voltage is halved, the new current $I'$ can be found using: $I' = \frac{V/2}{R} = \frac{I}{2} = \frac{15.87}{2} = 7.935 \text{ A}$

4. Part (C) – One-third Current and New Voltage: If the current is reduced to one-third of the initial current, the new voltage $V'$ can be calculated as: $V' = I' \times R = \left(\frac{I}{3}\right) R = \frac{V}{3}$

To finalize the answers, we need either the initial voltage or additional context about the potential difference between points A and B. If you have any further details on the voltage or would like assistance with specific formulas, let me know!

Would you like more details on each step? Here are some related questions to deepen understanding:

1. How can we determine the potential difference between two points if only distances are given?
2. What is Ohm’s Law and how does it relate to current, voltage, and resistance?
3. How does changing the voltage in a circuit affect the current?
4. How would we calculate the electric field between points A and B?
5. What effect does resistance have on the overall current in a circuit?

Tip: In physics and circuit problems, it's essential to identify the known variables first and check if all necessary values for calculations are available.